
5-dimensional algebras with identity \(xyz=0\)  (also known as 2-step nilpotent algebras) is the intersection of all varieties of algebras defined by a family of polynomial identities of degree three or more; for example, it is in intersection of associative, Zinbiel, Leibniz, etc, algebras. All these algebras can be obtained as central extensions of zero-product algebras. The geometric classification of 2-step nilpotent algebras is given in [1]}. It is the reason why we are not interested in it.

5-dimensional Zinbiel (non-2-step nilpotent) algebras, which are central extensions of Zinbiel algebras with nonzero product of a smaller dimension. These algebras are classified by several steps:

split complex 5-dimensional Zinbiel algebras are classified in [2]}, [3]};

non-split complex 5-dimensional Zinbiel algebras with 2-dimensional annihilator are classified in [3]};

non-split complex 5-dimensional Zinbiel algebras with 1-dimensional annihilator are classified in Theorem A (see below).


